91影视

The width of the crack between two walls is $10\text{鈥塩尘}$.The speed of the electron and the baseball is $50\text{鈥尘/蝉}$, and the weight of the baseball is$0.145\text{鈥塳驳}$ .

It is known that the diffraction minima occur at $m位=Dsin胃$.

Here m is integer and D is the slit width, $\mathbf{胃}$is the angle made by the incident wave with the normal and$\mathbf{位}$ is wave length of the incident wave.

(a)

The diffraction minima for a single slit have the relation,

$m位=D\sin 胃$.

The value of is $1$,so, the full width can be obtained using the formula:

$螖胃=2{\sin }^{鈭�1}\left(\frac{位}{D}\right)$

Now, substitute $h=6.63脳{10}^{鈭�34}{\text{鈥尘}}^{\text{2}}鈰�\text{kg/s}$and $m=9.11脳{10}^{鈭�31}\text{鈥塳驳}$for electron and $v=50\text{鈥尘/蝉}$to find the value of $位$by using the de-Broglie relation as:

$\begin{array}{c}位=\frac{h}{mv}\\ =\frac{6.63脳{10}^{鈭�34}{\text{鈥尘}}^{\text{2}}鈰�\text{kg/s}}{9.11脳{10}^{鈭�31}\text{鈥塳驳}脳50\text{鈥尘/蝉}}\\ =1.46脳{10}^{鈭�5}\text{鈥尘}\end{array}$

Substitute $位=1.46脳{10}^{鈭�5}\text{鈥尘}$and$D=10\text{鈥塩尘}=0.1\text{鈥尘}$ to find the spreading as follows:

$\begin{array}{c}螖胃=2{\sin }^{鈭�1}\left(\frac{位}{D}\right)\\ =2{\sin }^{鈭�1}\left(\frac{1.46脳{10}^{鈭�5}\text{鈥尘}}{0.1\text{鈥尘}}\right)\\ =2脳0.00835掳\\ =0.0167掳\end{array}$

In this case, the electron is able to produce a diffraction pattern that is at a very small angle but measurable.

Thus, The angular width of the central diffraction maximum when an electron passes through is $0.0167掳$.

(b)

Substitute $h=6.63脳{10}^{鈭�34}{\text{鈥尘}}^{\text{2}}鈰�\text{kg/s}$and $m=0.145\text{鈥塳驳}$for electron and $v=50\text{鈥尘/蝉}$to find the value of $位$by using the de-Broglie relation:

$\begin{array}{c}位=\frac{h}{mv}\\ =\frac{6.63脳{10}^{鈭�34}{\text{鈥尘}}^{\text{2}}鈰�\text{kg/s}}{0.145\text{鈥塳驳}脳50\text{鈥尘/蝉}}\\ =9.14脳{10}^{鈭�35}\text{鈥尘}\end{array}$

Substitute $位=9.14脳{10}^{鈭�35}\text{鈥尘}$and$D=10\text{鈥塩尘}=0.1\text{鈥尘}$ to find the spreading as follows:

$\begin{array}{c}螖胃=2{\sin }^{鈭�1}\left(\frac{位}{D}\right)\\ =2脳{\sin }^{鈭�1}\left(\frac{9.14脳{10}^{鈭�35}\text{鈥尘}}{0.1\text{鈥尘}}\right)\\ =2脳5.24脳{10}^{鈭�32}掳\\ =1.05脳{10}^{鈭�31}掳\end{array}$

This is a very small angle. Even Labs won鈥檛 be able to detect this interference pattern.

Thus, the angular width of the central diffraction maximum when baseball passes through is$1.05脳{10}^{鈭�31}掳$ .

(c)

Le the particle directed along the $x$-direction with the slit opening along the$y$-direction. This wave confinement along $y$-direction will produce uncertainty in the momentum $螖p_y$. We have to find the uncertainty in the $y$-component of momentum using the Heisenberg relation.

The first case is of the electron:

$\begin{array}{c}螖y螖p_y鈮�\frac{\mathrm{鈩�}}{2}\\ 螖p_y鈮�\frac{\mathrm{鈩�}}{2螖y}\\ 螖p_y鈮�\frac{1.054脳{10}^{鈭�34}{\text{鈥尘}}^{\text{2}}鈰�\text{kg/s}}{2脳0.1\text{鈥尘}}\\ =5.27脳{10}^{鈭�34}\text{鈥塳驳}鈰�\text{m/s}\end{array}$

The initial momentum can be caculated as,

$\begin{array}{c}{p}_i=p_x\\ =mv\\ =9.11脳{10}^{鈭�31}\text{鈥塳驳}脳50\text{鈥尘/蝉}\\ =4.55脳{10}^{鈭�29}\text{鈥塳驳}鈰�\text{m/s}\end{array}$

So, the value of $\frac{螖p_y}{p_i}$can be caculated as,

$\begin{array}{c}\frac{螖p_y}{p_i}=\frac{5.27脳{10}^{鈭�34}\text{鈥塳驳}鈰�\text{m/s}}{4.55脳{10}^{鈭�29}\text{鈥塳驳}鈰�\text{m/s}}\\ =1.16脳{10}^{鈭�5}\end{array}$

The second case for baseball:

$\begin{array}{c}螖y螖p_y鈮�\frac{\mathrm{鈩�}}{2}\\ 螖p_y鈮�\frac{\mathrm{鈩�}}{2螖y}\\ 螖p_y鈮�\frac{1.054脳{10}^{鈭�34}{\text{鈥尘}}^{\text{2}}鈰�\text{kg/s}}{2脳0.1\text{鈥尘}}\\ =5.27脳{10}^{鈭�34}\text{鈥塳驳}鈰�\text{m/s}\end{array}$

The initial momentum can be caculated as,

$\begin{array}{c}{p}_i=p_x\\ =mv\\ =0.145\text{鈥塳驳}脳50\text{鈥尘/蝉}\\ =7.25\text{鈥塳驳}鈰�\text{m/s}\end{array}$

So, the value of $\frac{螖p_y}{p_i}$can be caculated as,

$\begin{array}{c}\frac{螖p_y}{p_i}=\frac{5.27脳{10}^{鈭�34}\text{鈥塳驳}鈰�\text{m/s}}{7.25\text{鈥塳驳}鈰�\text{m/s}}\\ =7.27脳{10}^{鈭�35}\end{array}$

From the above calculation, the ratio of uncertainty and the initial momentum is small but measurable in the case of the electron. Still, the ratio is very small and negligible in calculations.

Thus, the longer the wavelength of the object, the higher the disturbance associated with it (whether angular dispersion or momentum uncertainty).